Showing papers in "Mathematical Control and Related Fields in 2011"
TL;DR: In this paper, the authors summarize recent results on the controllability of parabolic systems of (several) parabolic equations, and the emphasis is placed on the extension of the Kalman rank condition (for finite dimensional systems of differing differential equations) to parabolic system.
Abstract: This paper tries to summarize recent results on the controllability of
systems of (several) parabolic equations. The emphasis is placed on the
extension of the Kalman rank condition (for finite dimensional systems of
differential equations) to parabolic systems. This question is itself tied
with the proof of global Carleman estimates for systems and leads to a wide
field of open problems.
143 citations
TL;DR: For families of partial differential equations (PDEs) with particular boundary conditions, strict Lyapunov functions are constructed in this article, which are used to prove asymptotic stability in the framework of an appropriate topology.
Abstract: For families of partial differential equations (PDEs) with particular boundary conditions, strict Lyapunov functions are constructed. The PDEs under consideration are parabolic and, in addition to the diffusion term, may contain a nonlinear source term plus a convection term. The boundary conditions may be either the classical Dirichlet conditions, or the Neumann boundary conditions or a periodic one. The constructions rely on the knowledge of weak Lyapunov functions for the nonlinear source term. The strict Lyapunov functions are used to prove asymptotic stability in the framework of an appropriate topology. Moreover, when an uncertainty is considered, our construction of a strict Lyapunov function makes it possible to establish some robustness properties of Input-to-State Stability (ISS) type.
127 citations
TL;DR: In this article, an inverse problem of determining a spatially varying function of the source by final overdetermining data is discussed. But the inverse problem is well-posed in the Hadamard sense except for a discrete set of values of diffusion constants.
Abstract: For a time fractional diffusion equation with
source term, we discuss an inverse problem of
determining a spatially varying function of the source by
final overdetermining data.
We prove that this inverse problem is well-posed
in the Hadamard sense except for a discrete set of values of
diffusion constants.
91 citations
TL;DR: In this article, a time-consistent optimal control problem is formulated for a controlled linear ordinary differential equation with a quadratic cost functional, and an equilibrium strategy based on a study of multi-person hierarchical differential games is presented.
Abstract: A time-inconsistent optimal control problem is formulated and
studied for a controlled linear ordinary differential equation with
a quadratic cost functional. A notion of time-consistent equilibrium
strategy is introduced for the original time-inconsistent problem.
Under certain conditions, we construct an equilibrium strategy which
can be represented via a Riccati--Volterra integral equation system.
Our approach is based on a study of multi-person hierarchical
differential games.
49 citations
TL;DR: In this paper, a method of time-delayed boundary feedback stabilization is proposed to stabilize the isothermal Euler equations locally around a given stationary subcritical state on a finite time interval.
Abstract: We consider the isothermal Euler equations with friction that model
the gas flow through pipes. We present a method of time-delayed
boundary feedback stabilization to stabilize the isothermal Euler
equations locally around a given stationary subcritical state on a
finite time interval. The considered control system is a quasilinear
hyperbolic system with a source term. For this system we introduce a
Lyapunov function with delay terms and develop time-delayed boundary
controls for which the Lyapunov function decays exponentially with
time. We present the stabilization method for a single gas pipe and
for a star-shaped network of pipes.
40 citations
TL;DR: In this paper, a class of initial boundary value problems (IBVP) of the Korteweg-de Vries equation posed on a finite interval with nonhomogeneous boundary conditions is studied and it is shown that the solutions exist globally as long as their initial value and the associated boundary data are small.
Abstract: In this paper, we study a class of initial boundary value problem
(IBVP) of the Korteweg-de Vries equation posed on a finite interval
with nonhomogeneous boundary conditions. The IBVP is known to be
locally well-posed, but its global $L^2$- a priori estimate
is not available and therefore it is not clear whether its solutions
exist globally or blow up in finite time. It is shown in this paper
that the solutions exist globally as long as their initial value and
the associated boundary data are small, and moreover, those
solutions decay exponentially if their boundary data decay
exponentially.
32 citations
TL;DR: In this article, the authors explain how transmutation techniques can be applied to derive observability results for the heat equation without any geometric restriction on the subset in which the control is being applied, from a good understanding of the wave equation.
Abstract: The goal of this note is to explain how transmutation techniques (originally introduced in [14] in the context of the control of the heat equation, inspired on the classical Kannai transform, and recently revisited in [4] and adapted to deal with observability problems) can be applied to derive observability results for the heat equation without any geometric restriction on the subset in which the control is being applied, from a good understanding of the wave equation. Our arguments are based on the recent results in [15] on the frequency depending observability inequalities for waves without geometric restrictions, an iteration argument recently developed in [13] and the new representation formulas in [4] allowing to make a link between heat and wave trajectories.
30 citations
TL;DR: In this article, a mathematical study of stability and controllability of one-dimensional network of ferromagnetic particles is presented, where the control is the magnetic field generated by a dipole whose position and amplitude can be selected.
Abstract: In this work,we present a mathematical study of stability and controllability of one-dimensional network of ferromagnetic particles. The control is the magnetic field generated by a dipole whose position and whose amplitude can be selected. The evolution of the magnetic field in the network of particules is described by the Landau-Lifschitz equation. First, we model a network of ellipsoidal shape ferromagnetic particles. Then, we prove the stability of relevant configurations and discuss the controllability by the means of the external magnetic field induced by the magnetic dipole. Finally some numerical results illustrate the stability and the controllability results.
25 citations
Metz1
TL;DR: In this paper, stability properties of indirectly damped systems of evolution equations in Hilbert spaces were investigated under new compatibility assumptions, and polynomial decay for the energy of solutions were obtained.
Abstract: We investigate stability properties of indirectly damped systems of evolution equations in Hilbert spaces, under new compatibility assumptions. We prove polynomial decay for the energy of solutions and optimize our results by interpolation techniques, obtaining a full range of power-like decay rates. In particular, we give explicit estimates with respect to the initial data. We discuss several applications to hyperbolic systems with hybrid boundary con- ditions, including the coupling of two wave equations subject to Dirichlet and Robin type boundary conditions, respectively.
24 citations
TL;DR: In this paper, the Lipschitz stability of the inverse problem for the wave equation with potential on a starshaped network was proved, and the main tool, proved in this paper, is a global Carleman estimate for the network.
Abstract: We are interested in an inverse problem for the wave equation with potential on a starshaped network. We prove the Lipschitz stability of the inverse problem consisting in the determination of the potential on each string of the network with Neumann boundary measurements at all but one external vertices. Our main tool, proved in this article, is a global Carleman estimate for the network.
23 citations
TL;DR: In this article, the controllability of a multilayer plate system with free boundary conditions was investigated and Carleman estimates were obtained by the method of Carlemen estimates.
Abstract: Exact controllability of a multilayer plate system
with free boundary conditions are
obtained by the method of Carleman estimates. The multilayer plate system
is a natural multilayer generalization of a three-layer "sandwich
plate'' system due to Rao and Nakra. In the multilayer version, $m$
shear deformable layers alternate with $m+1$ layers modeled under
Kirchoff plate assumptions. The resulting system involves $m+1$
Lame systems coupled with a scalar Kirchhoff plate equation. The
controls are taken to be distributed in a neighborhood of the boundary.
This paper is the sequel to [2] in which only clamped and hinged boundary conditions are considered.
TL;DR: In this paper, a vector-host epidemic model with control measures is considered to assess the impact of control measures on the prevalence of the vector host diseases, and the authors provided sufficient conditions for the local stability of the disease free equilibrium and the sensitivity analysis for the reproduction number with respect to the model parameters.
Abstract: In this paper, a vector-host epidemic model with control measures is considered to assess the impact of control measures on the prevalence of the vector-host diseases. We incorporated mosquito-reduction strategy and host medical treatment into the model. For the basic vector-host model, we provide sufficient conditions for the local stability of the disease free equilibrium (DFE) and the sensitivity analysis for the reproduction number with respect to the model parameters. Using the optimal control theory, the optimal levels of the two controls are characterized, and then the existence and uniqueness for the optimal control pair are established. Numerical simulations are further conducted to confirm and extend the analytical results. Numerical results suggest that optimal multi-control strategy is a more beneficial choice in fighting the outbreak of vector-host diseases. For the vector-host epidemics, vector control measures should be taken prior to other measures.
TL;DR: In this paper, the authors developed numerical methods for finding optimal dividend policies to maximize the expected present value of the dividend payment, where the surplus follows a regime switching jump and the switching is represented by a continuous-time Markov chain.
Abstract: This work develops numerical methods for finding optimal dividend
policies to maximize the expected present value of dividend
payout, where the surplus follows a regime-switching jump
diffusion model and the switching is represented by a
continuous-time Markov chain To approximate the optimal dividend
policies or optimal controls, we use Markov chain approximation
techniques to construct a discrete-time controlled Markov chain
with two components Under simple conditions, we prove the
convergence of the approximation sequence to the surplus process
and the convergence of the approximation to the value function
Several examples are provided to demonstrate the performance of
the algorithms
TL;DR: In this paper, a Mindlin-Timoshenko model with non constant coefficients and non smooth coefficients set in a bounded domain is proposed, where the model corresponds to the coupling between the wave equation and the dynamical elastic system.
Abstract: A
Mindlin-Timoshenko model with non constant
and non smooth coefficients set in a bounded domain
of $\mathbb{R}^d, d\geq 1$ with some internal dissipations is proposed.
It corresponds to the coupling between the wave
equation and the dynamical elastic system.
If the dissipation acts on both equations,
we show an exponential decay rate.
On the contrary if the dissipation is only active on the elasticity equation, a
polynomial decay is shown; a similar result is proved in one dimension if the dissipation is only active on the wave equation.
TL;DR: In this article, the authors studied the internal stabilization of a coupled system of generalized Korteweg-de Vries equations under the effect of a localized damping term and proved the unique continuation property for weak solutions.
Abstract: The purpose of this work is to study the internal stabilization
of a coupled system of two generalized Korteweg-de Vries equations under the
effect of a localized damping term. The exponential stability, as well as,
the global existence of weak solutions are investigated when the exponent in
the nonlinear term ranges over the interval $[1, 4)$. To obtain the decay we use multiplier techniques
combined with compactness arguments and reduce the problem to prove a unique continuation property
for weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.
TL;DR: In this article, the null controllability of Oseen and Navier-Stokes type systems was shown to be uniform in terms of the penalization parameter, and a null control with a uniformly bounded $L^2$ norm was provided.
Abstract: In this note we use the result of [22] to prove a global Carleman inequality related to the null controllability of penalized Stokes kind systems. The constants of the obtained Carleman inequality are uniform in terms of the penalization parameter $\varepsilon$. It then provides a null control with a uniformly (in $\varepsilon$) bounded $L^2$ norm. With a limiting argument we also deduce a new Carleman inequality for Stokes type system. Thus, we apply theses results to obtain the null controllability of Oseen and Navier-Stokes system in the penalized and in the non penalized cases.
TL;DR: In this article, the authors prove some decay estimates of the energy of the wave equation governed by localized nonlinear dissipations in a bounded domain in which trapped rays may occur, based on a comparison with the linear damped wave equation and an interpolation argument.
Abstract: We prove some decay estimates of the energy of the wave equation
governed by localized nonlinear dissipations in a bounded domain in
which trapped rays may occur. The approach is based on a comparison
with the linear damped wave equation and an interpolation argument.
Our result extends to the nonlinear damped wave equation the
well-known optimal logarithmic decay rate for the linear damped wave
equation with regular initial data.
TL;DR: In this paper, a pathwise stochastic Taylor expansion is defined around any random time-space point (τ,ξ), where the temporal component τ does not even have to be a stopping time.
Abstract: In this paper we study the pathwise stochastic Taylor expansion, in
the sense of our previous work [3], for a class of
Ito-type random fields in which the diffusion part is allowed to contain
both the random field itself and its spatial derivatives.
Random fields of such an "self-exciting" type particularly contains the
fully nonlinear stochastic PDEs of curvature driven diffusion, as well as
certain stochastic Hamilton-Jacobi-Bellman equations. We introduce the new notion
of "$n$-fold" derivatives of a random field, as a fundamental device to cope
with the special self-exciting nature. Unlike our previous work [3],
our new expansion can be defined around any random time-space point (τ,ξ), where
the temporal component τ does not even have to be a stopping time. Moreover, the exceptional
null set is independent of the choice of the random point
(τ,ξ). As an application, we show how this new form of pathwise Taylor expansion
could lead to a different treatment of the stochastic characteristics for a class of fully
nonlinear SPDEs whose diffusion term involves both the solution and its gradient, and hence
lead to a definition of the stochastic viscosity solution for such SPDEs, which is new
in the literature, and potentially of essential importance in stochastic control theory.
TL;DR: In this article, an optimal control problem governed by a semilinear elliptic partial differential equation is considered and an existence result is established under a Cesari-type condition.
Abstract: An optimal control problem governed by semilinear elliptic partial
differential equation is considered. The equation is in divergence
form with the leading term containing controls. By studying the
$G$-closure of the leading term, an existence result is established
under a Cesari-type condition.
Journal Article•
TL;DR: In this article, a class of initial boundary value problems (IBVP) is studied and it is shown that the solutions exist globally as long as their initial value and the associated boundary data are small and moreover, those solutions decay exponentially if their boundary data decay exponentially.
Abstract: In this paper, we study a class of initial boundary value problem (IBVP) of the Korteweg-de Vries equation posed on a finite interval with nonhomogeneous boundary conditions. The IBVP is known to be locally wellposed, but its global L2− a priori estimate is not available and therefore it is not clear whether its solutions exist globally or blow up in finite time. It is shown in this paper that the solutions exist globally as long as their initial value and the associated boundary data are small, and moreover, those solutions decay exponentially if their boundary data decay exponentially
TL;DR: In this paper, the authors explore, in both continuous and discrete settings, some methods for approximate solution of such identification problems in a one dimensional linear elasticity framework, for which they suggest the names adjoint null space method or complementary projection method.
Abstract: The determination of parameter distributions, including fault detection, in elastic structures is a subject of great importance in structural engineering and related areas of applied mathematics. In this article we explore, in both continuous and discrete settings, some methods for approximate solution of such identification problems in a one dimensional linear elasticity framework. Methods for related optimization problems based on the matrix trace norm are described. The main objective of the paper is to introduce a method, believed new with this article, for which we suggest the names adjoint null space method or complementary projection method. Computational results for sample problems based on this technique are presented.
TL;DR: For the semi-linear problem, this article showed that there exists a unique solution tending to at least ε(1+t) √ ε − ε, where ε is the Lipschitz constant.
Abstract: If $L$ is the generator of a uniformly bounded group of operators $T(t)$ on a Banach space $X$, the abstract evolution
equation $ u' + Lu(t) = h(t) $ has a (weak) solution tending to $0$ as
$t\rightarrow +\infty $ if, and only if $\int_0^{+\infty}T(s) h(s) ds $ is semi-convergent, and then this solution is unique.
For the semi-linear equation $ u' + Lu(t) + f(u) = h(t) $, if $f$ such that $f(0) = 0$ is Lipschitz continuous on bounded subsets
of $X$ and has a Lipschitz constant bounded by
$ Cr^\alpha $ in the ball $B(0, r)$ for $r\leq r_0$, for any $h$ satisfiying
$||h(t)|| \leq c(1+t)^{-(1+ \lambda )} $
with $\lambda >\frac{1}{\alpha}$ and $c$ small enough
there exists a unique solution tending to $0$ at least like $(1+t)^{- \lambda}.$ When the system is dissipative,
this special solution makes it sometimes possible to estimate from below the rate of decay to $0$ of the other solutions.
TL;DR: In this article, the authors considered the blow-up solutions of the Cauchy problem for the critical nonlinear Schrodinger equation with a repulsive imbalance of harmonic potential.
Abstract: We consider the blow-up solutions of the Cauchy problem for the
critical nonlinear Schrodinger equation with a repulsive
harmonic potential. In terms of Merle and Tsutsumi's arguments as
well as Carles' transform, the $L^2$-concentration property of
radially symmetric blow-up solutions is obtained.